If a number is exactly Divisible by 85, then what will be the Remainder when the same number is Divided by 17?

If a number is exactly Divisible by 85, then what will be the Remainder when the same number is Divided by 17?

To find the remainder when a number that is exactly divisible by 85 is divided by 17, follow these steps

Let the number be 85n, where n is a whole integer. This means the number is a multiple of 85.

Now, divide 85n by 17

85n/17 = 5n

Since n is a whole integer, 5n will also be a whole integer. Therefore, 5n represents the quotient when 85n is divided by 17.

According to the division theorem in mathematics, the remainder when a number is divided by 17 is the difference between the number and the largest multiple of 17 that is less than or equal to the number. Since 85n is exactly divisible by 17, the remainder will be 0

Therefore, when a number is exactly divisible by 85, the remainder when the same number is divided by 17 is 0.

Concept of Divisibility and Remainders

Divisibility is a fundamental concept in mathematics that determines whether one number can be divided evenly by another without leaving a remainder. For example, a number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.

Testing Divisibility:

Mathematicians use rules and tests to determine if one number divides another without leaving a remainder. These rules often involve properties of numbers, such as their digits or factors, and provide efficient ways to check divisibility.

Importance in Number Theory:

Divisibility plays a crucial role in number theory, a branch of mathematics that studies the properties and relationships of numbers. It helps in classifying numbers into different categories based on their divisors and aids in solving problems related to factors and multiples.

Finding Remainders:

When one number is divided by another and does not divide evenly, the leftover part is called the remainder.

Modular Arithmetic:

The study of remainders, or residues, is central to modular arithmetic. This branch of mathematics deals with operations on remainders and their properties, which find applications in cryptography, computer science, and theoretical physics.

Applications in Problem Solving:

Divisibility and remainders are used extensively in problem-solving across various fields. They help in determining factors of large numbers, checking divisibility rules in calculations, and solving real-world problems involving distribution and allocation. Learning about divisibility rules and remainders enhances mathematical reasoning and problem-solving skills. It strengthens understanding of number properties and prepares students for advanced mathematical concepts.